In number theory, the BatemanâHorn conjecture is a statement concerning the frequency of prime numbers among the values of a system of polynomials, named after mathematicians Paul T. Bateman and Roger A. Horn who proposed it in 1962. It provides a vast generalization of such conjectures as the Hardy and Littlewood conjecture on the density of twin primes or their conjecture on primes of the form n2 + 1; it is also a strengthening of Schinzel's hypothesis H.
The BatemanâHorn conjecture provides a conjectured density for the positive integers at which a given set of polynomials all have prime values. For a set of m distinct irreducible polynomials Æ1, ..., Æm with integer coefficients, an obvious necessary condition for the polynomials to simultaneously generate prime values infinitely often is that they satisfy Bunyakovsky's property, that there does not exist a prime number p that divides their product f(n) for every positive integer n. For, if there were such a prime p, having all values of the polynomials simultaneously prime for a given n would imply that at least one of them must be equal to p, which can only happen for finitely many values of n or there would be a polynomial with infinitely many roots, whereas the conjecture is how to give conditions where the values are simultaneously prime for infinitely many n.
An integer n is prime-generating for the given system of polynomials if every polynomial Æi(n) produces a prime number when given n as its argument. If P(x) is the number of prime-generating integers among the positive integers less than x, then the BatemanâHorn conjecture states that
where D is the product of the degrees of the polynomials and where C is the product over primes p
with the number of solutions to
Bunyakovsky's property implies for all primes p, so each factor in the infinite product C is positive. Intuitively one then naturally expects that the constant C is itself positive, and with some work this can be proved. (Work is needed since some infinite products of positive numbers equal zero.)
As stated above, the conjecture is not true: the single polynomial Æ1(x) = −x produces only negative numbers when given a positive argument, so the fraction of prime numbers among its values is always zero. There are two equally valid ways of refining the conjecture to avoid this difficulty:
It is reasonable to allow negative numbers to count as primes as a step towards formulating more general conjectures that apply to other systems of numbers than the integers, but at the same time it is easy to just negate the polynomials if necessary to reduce to the case where the leading coefficients are positive.
If the system of polynomials consists of the single polynomial Æ1(x) = x, then the values n for which Æ1(n) is prime are themselves the prime numbers, and the conjecture becomes a restatement of the prime number theorem.
If the system of polynomials consists of the two polynomials Æ1(x) = x and Æ2(x) = x + 2, then the values of n for which both Æ1(n) and Æ2(n) are prime are just the smaller of the two primes in every pair of twin primes. In this case, the BatemanâHorn conjecture reduces to the HardyâLittlewood conjecture on the density of twin primes, according to which the number of twin prime pairs less than x is
When the integers are replaced by the polynomial ring Fu] for a finite field F, one can ask how often a finite set of polynomials fi(x) in Fu][x] simultaneously takes irreducible values in Fu] when we substitute for x elements of Fu]. Well-known analogies between integers and Fu] suggest an analogue of the BatemanâHorn conjecture over Fu], but the analogue is wrong. For example, data suggest that the polynomial
in F3u][x] takes (asymptotically) the expected number of irreducible values when x runs over polynomials in F3u] of odd degree, but it appears to take (asymptotically) twice as many irreducible values as expected when x runs over polynomials of degree that is 2 mod 4, while it (provably) takes no irreducible values at all when x runs over nonconstant polynomials with degree that is a multiple of 4. An analogue of the BatemanâHorn conjecture over Fu] which fits numerical data uses an additional factor in the asymptotics which depends on the value of d mod 4, where d is the degree of the polynomials in Fu] over which x is sampled.